Ahh, the first thing I'm going to say is, no matter what it is you decide to watch, make sure you watch it together. Gauge her interest and try a few different things. But as Kel said, comedy is the easiest way in.
For movie specifics -
Ponyo
Howl's Moving Castle
My Neighbor Totoro
These were all made by Studio Ghibli, same people that made Princess Mononoke and Spirited Away. They all have that childish charm and lovely art styles that you just can't get anywhere else. Japanese Disney is really the best way to describe them.
For series specifics -
Chobits -
Gabrinarra loved this one. It's adorable, and it's in the direction of the future without being too "OMG FUTURE".
Ranma 1/2 - Funny, witty, martial arts, love story, AMAZING characters, it has it all. This was one of my first introductions with Caroline, and it really helped with making that connection. There are some boobs in it, but it's not crazy like current anime.
Sword Art Online - If she has any experience in the MMO world, you guys should give it a shot together. I'm all about the love story in this series. It is a bit actiony and a little over the top in terms of original premise, but she might enjoy it. I know Caroline did.
Tenchi Muyo / Tenchi Universe - It's technically a Harem anime, but it's actually pretty damn funny. I personally prefer Muyo, but my buddies all enjoy Universe more. And we don't talk about Tenchi in Tokyo.
Shuffle! - This one starts off kinda cute, but turns into a bit of a psychological drama. May be a bit rough as an entry level piece, but may show that anime is a bit more deep than it comes off to be.
These should be a good start and a bit of a few different styles without being over the top.